In this Thesis we study topologies on countable sets from the perspective of Tukey reductions of their neighbourhood filters. In particular we will study $k$-analytic group topologies on $\omega$. This will allow us to obtain a metrization theorem for analytic sequential group topologies on $\omega$, as well as a classification of such groups in terms of the Tukey type of the filter of neighbourhoods of the identity. Recall that a countable topological space is analytic if the topology is analytic as a subset of the Cantor space.